George J. Rebane
2.1 Field of Invention
The invention relates generally to the field of methods and software products for financial analysis risk management, and more particularly to methods and software products for investment portfolio design and the selection, analysis of investments and the allocation of investment assets among investments.
2.2 Background of the Invention
2.2.1 CAPM and CML Background
In this section we present sufficient background of the Capital Asset Pricing Model (CAPM) and the Capital Market Line (CML) to establish the departure points for derivation of the present invention: Risk Direct Asset Allocation (RDAA) and Risk Resolved CAPM (RR/CAPM). A complete tutorial on modern asset allocation methods, particularly the CAPM and the related Arbitrage Pricing Theory, may be found in any one of a number of good texts on corporate finance [4] (A bibliography of reference is found at the end of this disclosure).
The practical application of any quantitative method of portfolio design based on securities"" covariance requires the selection of a xe2x80x98short listxe2x80x99 of N risky stocks or other securities. Several studies have shown that the investor begins to gain xe2x80x9calmost all the benefits of (portfolio) diversificationxe2x80x9d at Nxe2x89xa68, xe2x80x9cvirtually no risk reductionxe2x80x9d for N greater than 15 [14], and measurable liabilities increasing beyond N=30 [15]. The nomination of the short list may be approached as a formal problem in multi-attribute utility [1]. We proceed here with a specified candidate set of N risky securities whose singular utility to the investor is their ability to contribute to a successful portfolio design.
The motivation for going beyond the CAPM, with its ever-present companion query as to xe2x80x9cwhether variance is the proper proxy for riskxe2x80x9d [21], is in the answer that variance is only the progenitor of risk and not its final measure. Between the two there is a road, unique to each investor, to be traveled that lets us individually answer the question xe2x80x9chow much of each of the N securities should Ixe2x80x94not he and not shexe2x80x94buy and/or hold?xe2x80x9d This question is answered by the present invention.
2.2.1.1 Risk
In the CAPM risk is measured by the rate performance dispersion of a security as expressed by its historical rate standard deviation. A primary problem with the CAPM is that once established, this xe2x80x98sigmaxe2x80x99 is applied uniformly to all investors independent of the amount they intend to invest or their individual aversion to the possible loss of investment assets. Thus the CAPM has a very egalitarian view of risk, and treats all investors equally, regardless of their total investment assets available for investment and net worth. The levels of risk and the concordant performance of a set of risky securities are quantified by their covariance matrix usually computed from specified historical data.
Suppose we have a candidate portfolio of N risky securities Si, i=[1,N]. We select a past performance epoch TPE and compute the symmetrical Nxc3x97N covariance matrix [5] for the securities as
cov S=E{(sxe2x88x92xcexc)(sxe2x88x92xcexc)T}xe2x80x83xe2x80x83(1)
where s is the rate of return (column) N-vector of the securities and xcexc is the vector of mean or expected returns computed over a past epoch TPE where
xcexci=ŝi, i=1,N.xe2x80x83xe2x80x83(2)
We now allocate a portfolio fraction ƒi to each of the N securities with the elements of ƒ summing to one. The total rate of return variance of such a portfolio is then given by
"sgr"2s(ƒ)=ƒTcov Sƒxe2x80x83xe2x80x83(3)
which shows the dependence of the portfolio""s return variance on the allocation vector ƒ. In modern portfolio theory [4] it is "sgr"(ƒ) from (3) that gives the uniform measure of portfolio risk for all investors, and thus constrains CAPM to treat all investors equally.
The expected rate of return for each risky security over the investment horizon (Ti) is predicted on the basis of its beta (xcex2) computed with respect to xe2x80x98the marketxe2x80x99 (e.g. SandP500) as follows.                               β          i                =                                            σ                              i                ,                M                            2                                      σ              M              2                                =                                    cov              ⁡                              (                                                      s                    1                                    ,                                      R                    M                                                  )                                                    σ              M              2                                                          (        4        )            
The familiar beta is further represented as the slope of a straight line relationship between market variation and the security in question. The frequently omitted alpha (xcex1) parameter defines the intercept of the least squares regression line that best fits a set of security and market return rates. A method for predicting a stock""s price {circumflex over (R)}M from a prediction of market performance {circumflex over (R)}M over Ti then yields
ŝi=xcex1i+xcex2i{circumflex over (R)}Mxe2x80x83xe2x80x83(5)
The classical CAPM formula for ŝi [4] generates the Security Market Line                                                                                           s                  ^                                i                            =                              xe2x80x83                            ⁢                                                R                  RF                                +                                                      β                    i                                    xc3x97                                      (                                          historical                      ⁢                                              xe2x80x83                                            ⁢                      market                      ⁢                                              xe2x80x83                                            ⁢                      risk                      ⁢                                              xe2x80x83                                            ⁢                      premium                                        )                                                                                                                          =                              xe2x80x83                            ⁢                                                R                  RF                                +                                                      β                    i                                    ⁡                                      (                                                                                            R                          ^                                                M                                            -                                              R                        RF                                                              )                                                                                                          (        6        )            
where RRF is the current risk free lending rate (the historical market risk premium has been calculated at 8.5%) and {circumflex over (R)}M is the expected return on the market over the investment horizon.
Keeping in mind the ability here to use other predictive security return models, in the remainder we will use the more straightforward (6) for predicting the performance of a security and understand the quoted xe2x80x98sigmaxe2x80x99 (standard deviation) of such a security to derive from the regression fit of K points [5] over TPE.                               σ          i                =                                            1                              K                -                1                                      ⁢                                          ∑                                  k                  =                  1                                K                            ⁢                                                {                                                            s                      k                                        -                                          (                                                                        α                          i                                                +                                                                              β                            i                                                    ⁢                                                      R                                                          M                              ,                              k                                                                                                                          )                                                        }                                2                                                                        (        7        )            
Combining a security""s expected rate of return and its standard deviation then yields the needed parameters for its assumed probability density function (p.d.f.) which fully characterizes the performance of the individual security with respect to the specified future performance of the market {circumflex over (R)}M over Ti.
For the RR/CAPM and RDAA developments below we additionally acknowledge an uncertain future market and express this by its variance "sgr"M2 to reflect the dispersion about the predicted mean return {circumflex over (R)}M. This additional uncertainty will be reflected in a given security""s xe2x80x98sigmaxe2x80x99 to yield its total standard deviation as
"sgr"T,i={square root over ("sgr"i2+L +xcex2i2+L "sgr"M2+L )}xe2x80x83xe2x80x83(8)
2.2.1.2 The Feasible and Efficient Sets
From corporate finance texts [4] we learn that a set of points termed the feasible set can be represented in 2-space where expected portfolio return {circumflex over (R)}P is plotted (FIG. 1) against the standard deviation ss of the portfolio given in (3). The expected return of the xe2x80x98riskyxe2x80x99 portfolio allocated according to ƒ is simply                                           R            ^                    P                =                              ∑                          i              =              1                        N                    ⁢                                    f              t                        ⁢                                          s                ^                            i                                                          (        9        )            
The efficient set is defined as the upper boundary of the feasible set drawn upward from the xe2x80x98minimum variance pointxe2x80x99 (MVP) since it is not reasonable to choose portfolios with the lesser expected gains for the same xe2x80x98riskxe2x80x99 as measured by the portfolio""s "sgr"s. Therefore, according to the CAPM the optimal portfolios are all represented by the infinite set of optimal allocation vectors {ƒ*} that define this upper boundary. The CAPM proceeds to resolve the problem further by introducing the risk free lending option which gives rise to the Capital Market Line.
2.2.1.3 The Capital Market Line
As shown in FIG. 1, when we introduce the risk free lending option at rate RRF, we add the (N+1)th instrument and increase the dimension of the investor""s decision space to N. The CAPM argues that the optimum portfolio now lies along a linexe2x80x94the Capital Market Line (CML)xe2x80x94that originates from (0, RRF) and is tangent to the efficient set at some point E for which a unique ƒ* can be discovered. Selecting a point between (0, RRF) and E defines what fraction should be invested risk free with the remainder being invested pro rata at ƒ*. Points closer to E represent a larger fraction going into the risky portfolio of N stocks.
We note that the computation of the efficient set per se is not required for the solution of ƒ*. As seen from FIG. 1, it is clear that if the slope of the CML is maximized within the constraints that ƒ is a fraction vector whose elements sum to unity, then we would automatically obtain point E and the resulting CML. The needed slope is given by                               tan          ⁢                      xe2x80x83                    ⁢          θ                =                                                            R                E                            ⁡                              (                                  f                  _                                )                                      -                          R              RF                                                          σ              E                        ⁡                          (                              f                _                            )                                                          (        10        )            
where RE and "sgr"E are the coordinates of E which depend on ƒ. The optimal risky fraction is then obtained directly by solving the constrained non-linear optimization problem [7],[8].                                           f            _                    *                =                  arg          ⁢                      xe2x80x83                    ⁢                                    max                              f                _                                      ⁢                          [                                                                                          R                      E                                        ⁡                                          (                                              f                        _                                            )                                                        -                                      R                    R                                                                                        σ                    E                                    ⁡                                      (                                          f                      _                                        )                                                              ]                                                          (        11        )            
which yields RE*(ƒ*) and "sgr"E*(ƒ*) from (9) and (3) respectively.
The resulting (fractional) portfolio design ƒP is finally determined from
ƒP=[ƒRF,(1xe2x88x92ƒRF)ƒR*T]Txe2x80x83xe2x80x83(12)
by appropriately selecting ƒRF.
The Capital Market Line is presented as the efficient set of both risky and risk free investments and culminates the CAPM""s efforts at defining a portfolio by leaving the investor with yet another infinite set of options from which to choose. At this point the CAPM simply asks the investor to apply his/her own method for picking ƒRF, or as stated in [4]:
xe2x80x9cHer position in the riskless asset, that is, the choice of where on the (CML) line she wants to be, is determined by her internal characteristics, such as her ability to tolerate risk.xe2x80x9d
The CAPM offers no guidance of any analytical method for determining each investor""s allocation of investment assets on the CML.
We note that during the course of the CAPM solution there has been no discussion of actual cash amounts to be invested. The presumption being all along that, however finally obtained, the risky portfolio fractions ƒR would apply equally to billionaires and blue collar workers. This assumption thus fails to recognize that individual investors have distinct risk preferences that are intimately tied to their overall investment assets and net worth, and that as a result, would select different allocations of their investment assets.
Accordingly, it is desirable to provide a computer implemented method and software product that accounts for individual investor risk preferences as a function of the individual investor""s financial profile, and thereby determines for a given portfolio of investments (i.e. short list), the optimal allocation of the investor""s assets, or any portion thereof, among the investment assets.
The present invention, the Risk Direct Asset Allocation and Risk Resolved CAPM, overcomes the limitations of conventional portfolio design methods including the CAPM, and software products by determining for an individual investor that investor""s risk tolerance function and selecting a monetary allocation of investment assets according to both the risk tolerance function, and quantifiable risk dispersion characteristics of a given allocation of investment assets in the portfolio. Generally RDAA and RR/CAPM are based on integrating key elements of modern utility, securities"" performance prediction, and optimization theories (see, e.g., [1],[2],[3]) that relate to risk averse behavior in matters of monetary uncertainty.
3.1 Investor Utility and Probability Preference Curves
In accordance with one embodiment of the present invention, a risk tolerance function (xe2x80x9cRTFxe2x80x9d) of the individual investor is determined. The risk tolerance function describes the investor""s probability preferences at each of the number of monetary amounts relative to the investor""s total assets. More specifically, at a given monetary amount A, the risk tolerance function for an investor defines the probability PP(A) at which the investor is indifferent between 1) receiving the monetary amount A, or 2) accepting the risk or gamble of receiving an investor defined putative best amount AH (for xe2x80x98happinessxe2x80x99 representing monetary contentment at which net worth the investor is willing to suffer essentially zero risk for further increasing his net assets) with probability PP(A) or losing his monetary assets and ending up at an investor defined putative worst amount AD (for xe2x80x98despairxe2x80x99) with probability 1-PP(A). The amounts AD and AH enclose the investor""s total net current assets AT. Preferably all investment amounts and outcome calculations will be based on AT and appropriate changes to this value. Some investors may instead consider AT to be net investable assets or even their net worth. Overall then, the risk tolerance function quantitatively defines the investor""s risk aversion or risk seeking behavior with respect to his unique monetary range of specified monetary amounts. Thus, the risk tolerance function is specifically scoped to the investor""s actual and unique monetary range which includes his total investment assets so that it realistically quantifies the investor""s preferences with respect to potential outcomes effecting the investor""s assets, and hence usefully describes (i.e. quantifies probabilistically ) the investor""s preferences as to the market risk presented by various allocations of investment assets within a portfolio.
The investor""s risk tolerance function is derived interactively in a straightforward and systematic manner through a sequence of decisions involving so-called reference gambles. Examples of several risk tolerance functions for three different investors are shown in FIG. 2. In this figure, the normalized PP value aries between 0 and 1 as the monetary outcome ranges from the investor""s putative worst amount AD, to the amount of monetary contentment AH, such that PP(AD)=0 and PP(AH)=1. It is seen that the risk averse behaviors assumed here are represented by concave downward functions. The straight line joining PP(AD) and PP(AH) is the expected monetary value (EMV) line which characterizes the behavior of a risk neutral individual. Consequently the risk seeker""s curve lies below the EMV line and is concave upward.
We note that the different risk tolerance functions in FIG. 2 represent different individuals as indicated. The fact that one risk tolerance function, RTF3, goes into negative territory states that this investor is willing to assume some resulting debt as the worst monetary outcome of risky investment schemes. It is reasonable, though not necessary, to assume that most mature or older investors will be risk averse with AD greater than 0 such as in RTF1 and RTF2. All reasonable investors will exhibit AH greater than AT.
The monetary difference between the PP curve and EMV line at a given PP(AEMV) value is called the investor""s risk premium (RP) and is seen to be the amount the investor is willing to forego or pay in order to avoid the (fair) expected value gamble at PP(A). In the figure we see that all other asset parameters given equal, Investor #1 is more risk averse than Investor #2 since RP1 greater than RP2. Investor #3 appears to be a young person with little total assets who would be risk seeking soon after going into debt.
3.2 A General Overview of RR/CAPM and RDAA
For any given allocation of investment assets among investments in the portfolio, a probability density function can be determined which describes the rate performance dispersion of the portfolio""s predicted market performance. Conventionally, this probability density function is typically expressed with respect to a portfolio defined by fractional weightings of the investment assets, since CAPM is unable to distinguish between the risk preferences of different investors. In accordance with the present invention however, the probability density function of the portfolio""s predicted market performance is expressed with respect to the investor""s available investment assets, and more particularly, with respect to the investor""s risk tolerance function. Thus, this probability density function describes the dispersion of potential monetary gains and losses to the investor given a specific allocation of the investor""s investment assets among the portfolio. For a given probability density function, there is a mean or expected value of the probability density function. The probability density function of the portfolio, for example, describes the overall expected performance of the portfolio in monetary amounts.
In accordance with one aspect of the present invention, once the investor""s monetary risk tolerance function, and the probability density function of a given investment allocation are determined, it is possible to create a probability density function of the investor""s probability preferences with respect to the investor""s risk tolerance function. This probability density function expresses the dispersion of risk preferences that the investor would experience as a result of the investment allocation. The expected value of this probability density function of the investor""s probability preferences thus describes the overall risk preference of the investor for the specific monetary allocation of investment assets (as opposed to the conventional asset independent risk analysis).
In accordance with the present invention then, investment assets are allocated to the investments of the portfolio by maximizing the expected value of the probability density function of the investor""s probability preferences. The probability density function of the investor""s probability preferences is determined as a function of the probability density function of the portfolio""s predicted market performance with respect to the investment assets allocation policy and the investor""s risk tolerance function. The investment allocation that maximizes the expected value of the investor""s probability preferences best satisfies these preferences as they are defined by the investor""s risk tolerance function.
In contrast to conventional approaches, the investment allocation here describes the actual monetary amounts of the investment assets to be allocated to the investments of the portfolio. Further, because the investment allocation is determined with respect to the investor""s unique risk tolerance function(s), it accounts for the investor""s own particular asset base and their risk aversion or risk seeking behavior relative to such asset base. This contrasts with conventional methods that do not account for either the assets or the risk preferences of investors, and hence treat all investors as 1) having exactly the same assets; and/or 2) having exactly the same risk preferences and tolerances. For this reason, as shown above, conventional approaches based on the CAPM produce only an infinite set of potential allocations, leaving it up to the individual investor to arbitrarily allocate their actual investment assets from among the possible solutions along the CML.
The probability density function on the probability preference of the investor""s risk tolerance function may be determined in a variety of manners in accordance with the present invention. In one embodiment, this probability density function is determined by numerically mapping the probability density function of the portfolio with respect to the investment assets through the investor""s risk tolerance function and onto the probability preference axis. This embodiment is preferable where there is a significant probability of the investor""s total assets falling below AD, the despair amount. Such an outcome is typically predicated by a large rate standard deviation for the portfolio given the investment allocation. The allocation of investment assets amongst the portfolio investments is iteratively adjusted until the expected value of the probability density function on the probability preference axis is maximized. FIG. 3 illustrates an example of the mapping of the probability density function of a given portfolio allocation through an investor""s risk tolerance function onto the probability preference axis.
In an alternate embodiment, the expected value of the probability density function of the investor""s probability preferences is determined by direct computation. One method of direct computation is by solution of:                               E          ⁡                      (                          PP              |                              f                _                                      )                          =                              ∫                          -              ∞                        ∞                    ⁢                                    g              ⁡                              (                A                )                                      ⁢                          h              ⁡                              (                                  A                  |                                      f                    _                                                  )                                      ⁢                          xe2x80x83                        ⁢                          ⅆ              A                                                          (        13        )            
where:
g(A) is the investor""s risk tolerance function, g(A) xcex5[0, 1] for ADxe2x89xa6A less than AH, and g(AD)=0, and g(AH)=1;
AD is the investor defined putative worst monetary amount or xe2x80x98despairxe2x80x99 amount;
AH is the investor defined putative contentment monetary amount or xe2x80x98happinessxe2x80x99 amount; and,
h(A|ƒ) is the probability density function of the investment portfolio""s predicted performance with respect to the investor""s total assets given allocation policy ƒ.
The solution to (13) may be usefully approximated by a truncated Taylor series expansion of g(A), the investor""s risk tolerance function, about the expected value of h(A|ƒ). One such implementation resolves (13) to:                                           E            ^                    ⁡                      (                          PP              |                              f                _                                      )                          =                              g            ⁡                          [                                                μ                  A                                ⁡                                  (                                      f                    _                                    )                                            ]                                +                                    1              2                        ⁢                                          (                                                                            ⅆ                      2                                        ⁢                                          g                      ⁡                                              (                        A                        )                                                                                                  ⅆ                                          A                      2                                                                      )                                            μ                                  A                  ⁡                                      (                    f                    )                                                                        ⁢                                          σ                A                2                            ⁡                              (                                  f                  _                                )                                                                        (        14        )            
This form of the equation can be readily optimized over the selected securities for each investor to yield the actual monetary allocation over such securities to the investor""s maximum expected monetary probability preference.
An examination of (14) is particularly revealing with respect to asset allocation. The first r.h.s. term is simply a mapping of xcexcA onto the PP axis and is consistent with the fact that all sane RTFs are smoothly and monotonically increasing with A throughout their entire range. The second r.h.s term is of particular interest since it adjusts the expected value of the mapped cash distribution according to two factorsxe2x80x94the curvature of the risk tolerance function and the cash quantified standard deviation of the total portfolio both reflected in the xcexcA region of investor""s total assets.
We recall from FIG. 2 that risk aversion is represented by the RTF lying above the EMV line and thereby curving downward with increasing A. This translates to a negative value of the second derivative and means that a term proportional to "sgr"A2 is subtracted from the direct mapping of xcexcA through the RTF. We will refer to one half the RTF""s second derivative evaluated at xcexcA as the portfolio risk compensation coefficient (RCC). Therefore as we assume a portfolio design that increases its expected gain along the CML, we see that "sgr"A also increases. Since the RTF flattens out with increasing A, the RCC becomes less negative, but the increasing "sgr"A effect begins to dominate and the mapped mean, according to (14), reaches a maximum and begins decreasing at the optimal allocation point. The opposite occurs for risk seekers whose RTF falls below the EMV line in the xcexcA vicinity; here the RCC is positive and the risk compensation adds to or augments the directly mapped PP value of xcexcA. This rewards the investor in such a region of his anticipated total assets. Again, in the practical application of the invented algorithm and methodology to realistic short lists of stocks, the xe2x80x98risk seeking portfolioxe2x80x99 at a high RCC may be characterized by high variance being traded off against a low mean because the risk seeker fully expects the high variance to work for (not against) him. We presume that current portfolio designers can take comfort from this analysis since a directly evolved form of CAPM risk as defined in Modem Portfolio Theory is very much present in the new RR/CAPM method presented here, albeit expressed in monetary (not rate) terms and mapped into the conflict resolving preference probability space.
In accordance with the present invention, the foregoing analysis and computations are embodied in a software product for controlling and configuring a computer to receive data descriptive of various investments and their risk characteristics, to interactively determine an investor""s risk tolerance function, to allocate investment assets to an investment portfolio, to compute the probability density function of the portfolio""s performance with respect to the investor""s assets, and to compute and maximize the expected value of the probability density function of the investor""s probability preferences. Additionally, the present invention may also be used in a broader context as a monetary risk management tool to determine asset allocations among sectors (e.g. large cap, bonds, growth, value, technology, metals, and the like) and also to select among candidate projects (e.g. acquire XYZ Inc., 3introduce product line A vs. B, buy new production facility, and the like) in a corporate planning environment.
3.3 User Interface Features
In accordance with another aspect of the present invention, there are provided various user interfaces that graphically capture and represent the investment allocation of the investment assets, along with useful information describing portfolio performance. One user interface graphically displays for each investment in the portfolio the allocation of the investment assets to the selected securities in terms of both monetary and percentage allocations, along with user definable upper and lower bounds for the allocation. There is also displayed a graphical representation of the expected return of the portfolio given the investment allocation, preferably shown with a confidence interval.
The upper and lower bounds for each investment are dynamically manipulable, and can be adjusted by the user to change the range of potential allocations to the investment. As the user moves an upper or lower bound to allow an increase or decrease in the allocation, the overall investment allocation policy among the portfolio is automatically recomputed in order to again maximize the expected value of the probability density function of the investor""s probability preferences. This user interface thus allows the user to easily and dynamically manipulate the investment allocation and observe the impact of such allocations on the expected return of the portfolio.